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<b>let </b><font color="Maroon" title="c1">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/>





<b>thus </b><a NAME="E1:167"/>
( <font color="Maroon" title="c1">H</font> <a href="zf_lang.html#NR2" title="ZF_LANG:NR.2">is_membership</a>  implies <font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="zf_lang.html#NR2" title="ZF_LANG:NR.2">is_membership</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="167_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a><b> such that </b><a NAME="E2:167_1"/><i><font color="Green" title="E69">A1</font></i>: 
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c5">x2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D11" title="ZF_LANG:def.11">ZF_LANG:def 11</a><br/>

<a NAME="E3:167_1"/>
 ex <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> st <font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Olive" title="b2">z2</font>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E2:167_1"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="zf_lang1.html#T169" target="_self" title="ZF_LANG1:th.169">Th169</a></i>;<br/>
<b>hence </b><a NAME="E4:167_1"/>
<font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="zf_lang.html#NR2" title="ZF_LANG:NR.2">is_membership</a> 
 <b>by </b><i><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:167"/><i><font color="Green" title="E69">A2</font></i>: 
<font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="zf_lang.html#NR2" title="ZF_LANG:NR.2">is_membership</a> 
 ;<br/>

<a NAME="E3:167"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 &amp; 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#T29" title="ZF_LANG:th.29">ZF_LANG:29</a></i>;<br/>
<a NAME="E4:167"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:167"/><b>then </b><i><font color="Green" title="E70">A3</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">H</font> )
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E2:167"><i><font color="Green" title="E69">A2</font></i></a>, <a class="ref" href="zf_lang1.html#T148" target="_self" title="ZF_LANG1:th.148">Th148</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="zf_lang.html#T36" title="ZF_LANG:th.36">ZF_LANG:36</a></i>;<br/>
<a NAME="E6:167"/><b>then </b>
<font color="Maroon" title="c1">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#D4" target="_self" title="ZF_LANG1:def.4">Def4</a></i>;<br/>
<a NAME="E7:167"/><b>then </b>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E5:167"><i><font color="Green" title="E70">A3</font></i></a>, <a class="ref" href="zf_lang1.html#D4" target="_self" title="ZF_LANG1:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E8:167"/>
<font color="Maroon" title="c1">H</font> <a href="zf_lang.html#NR2" title="ZF_LANG:NR.2">is_membership</a> 
 <b>by </b><i><a class="ref" href="zf_lang.html#T42" title="ZF_LANG:th.42">ZF_LANG:42</a></i>;<br/>


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